The main of focus of this thesis is the study of cohomological symmetries. Namely, given an algebraic variety, we study the symmetries of its derived category, which are also known as autoequivalences. The thesis is split into five chapters. In §1 we give an introduction to the material presented in the thesis, as well as a motivation as to why one might be interested in studying these topics. We encourage the reader to have a look, so as to know what is coming. In §2, we set up the preliminary notions we will need throughout the whole thesis. The arguments touched in this chapter comprise triangulated categories, dg-categories, and spherical functors. In §3, we begin to present the novel mathematics developed in this thesis. The focus of this chapter is on how to compose spherical twists around spherical functors. We describe a general recipe that takes as input two spherical functors and outputs a new spherical functor whose twist is the composition of the twists around the functors we started with, and whose cotwist is a gluing of the cotwists. We conclude the chapter by specialising the theory to the case of spherical objects and P-objects. In §4, we study autoequivalences arising from geometric correspondences. We prove that such autoequivalences have a natural representation as the inverse of the spherical twist around a spherical functor, and that in some examples this geometric spherical functor agrees with the construction described in §3. We conclude the thesis with §5, in which we present some possible future applications of this work. In doing so, we hope to stimulate further mathematical discussion around topics that the author of this thesis finds really exciting.